Classical Least Squares, Multivariate Curve Resolution and Iterative Optimization Technology (CLS/MCR/IOT)

Course Description

Principal components analysis PCA) and inverse least squares (ILS) methods such as partial least squares (PLS) are ubiquitous to chemometrics. However, classical least squares (CLS or forward least squares) techniques are seeing a resurgence in popularity. CLS is a linear mixture model and is appropriate for systems that obey the multicomponent Beer’s law. Major reasons for the revival of CLS are better interpretability, the ability to control aspects of the regression modeling and it requires fewer calibration samples than ILS models. CLS also provides a natural framework for the development of popular de-cluttering methods such as External Parameter Orthogonalization (EPO) and Generalized Least Squares (GLS) weighting. These methods can be combined with CLS to form Gray CLS models which have greatly improved prediction performance compared to conventional CLS.

The CLS model is also the basis for Multivariate Curve Resolution (MCR) which is used to unmix unlabeled spectra (i.e. no concentrations) as pure components and contributions. Constraints, including non-negativity, closure and others, are used in MCR to obtain physically meaningful solutions. MCR can be used to create CLS models for use on future samples and for the development of chemical maps in from hyper spectral images.

CLS and MCR are both used with Iterative Optimization Technology (IOT) and Extended IOT (EIOT). IOT/EIOT has shown promise as a “lean chemometrics” method as it requires fewer calibration samples than PLS and other ILS methods.

This half-day course will start by covering CLS regression methods including classical, extended and Gray CLS. MCR and the use of constraints will be considered. The relationship between IOT/EIOT models and CLS and MCR will be elucidated. The course includes hands-on computer time using PLS_Toolbox or Solo to work through example problems.

Prerequisites

Linear Algebra for Machine Learning and Chemometrics, Chemometrics I –– PCA, and Chemometrics II – Regression and PLS or equivalent experience.

Course Outline

• Classical Least Squares (CLS) Models
  • Based on measured pure components
  • Identification from mixtures
  • Model application to new data
  • Model diagnostics
• Extensions to CLS
  • The Extended Mixture Model (EMM)
  • External Parameter Orthogonalization (EPO)
  • Generalized Least Squares (GLS)
  • Combining EPO/GLS and CLS: Gray CLS
• Multivariate Curve Resolution (MCR)
  • Identification with Alternating Least Squares (ALS)
  • Non-negativity constraints
  • Closure and other constraints
  • Pitfalls in MCR and how to avoid them
• Iterative Optimization Technology (IOT) and Extended IOT (EIOT)
  • The IOT idea: decreasing the calibration burden
  • Relationship to CLS, MCR and constraints
• Examples and Conclusions